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# Number Theory/Joint Proportionality

Question
QUESTION: Hello:

In one of my past question I asked how to solve the following:

4 carpenters can build 8 houses in 10 days.  2 carpenters can build how many houses in 15 days?

You provided this solution:

"we can use the idea of joint proportionality.  The number of houses built is jointly proportional to the number of carpenters and the number of days.  So h = kcd where k is a constant.  Substituting the given: 8 = k*40, so k = 1/5.  Our formula is then h = cd/5."

How was the idea of joint proportionality determined?  For example, how did you know to use h = kcd and not d = khc or c = khd?

If we double the number of carpenters, then we double the number of houses, so h is proportional to c.
If we double the number of days, we double the number of houses, so h is proportional to d.  So h is proportional to both c and d.  So it is proportional to the product.  This is the principle of joint variation.  Then h = kcd.
Hope that clarifies the problem.
Best wishes
vijilant

---------- FOLLOW-UP ----------

QUESTION: Hello:

I understand the calculation but not for all situations.  Here is another example:

24 men working 8 hours per day can do a piece of work in 15 days  In how many days can 20 men working at 9 hours per day do the same work?

If men are increased the time, hours and days, will decrease.  Time and days are indirectly proportional to men.  How is the solution determined for this example?

I thank you for your help and assistance.

Hello Kenneth

We are trying to find a number of days, so it is best to have d as the subject.
d is inversely (not indirectly) proportional to both men,m, and hours, h. So d = k/(mh).  Substituting the given gets us to k = 2880, and then using the formula gives the answer 16 days.

Best wishes
vijilant
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Number Theory

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#### Vijilant

##### Expertise

Most questions on number theory, divisibility, primes, Euclidean algorithm, Fermat`s theorem, Wilson`s theorem, factorisation, euclidean algorithm, diophantine equations, Chinese remainder theorem, group theory, congruences, continued fractions.

##### Experience

Teacher of math for 53 years

Organizations
AQA Doncaster Bridge Club Danum Strings Orchestra Doncaster Conservative Club Danum Strings Orchestra Simply Voices Choir Doncaster TNS mystery shopping St Paul's Music Group Cantley

Publications
Journal of mathematics and its applications M500 magazine

Education/Credentials
BSc (Hons) Liverpool (Science). BA (Hons) OU (Mathematics)

Awards and Honors
State Scholarship 1955 Highest Score in Yorkshire on OU course MST209 £50 prize First class honours in OU BA Mathematics

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I taught John Birt, former Director of the BBC in 1961. His homework book was the most perfect I have ever marked. And also the most neat. I could tell he was destined for great things. One of my classmates was the poet Roger McGough, and I have a mention in his autobiography.